To determine which function better models the world population in 1970 using the given models [tex]\( f(x) = 0.074x + 2.294 \)[/tex] and [tex]\( g(x) = 2.577 \times 1.017^x \)[/tex], we need to follow these steps:
### Step 1: Identify the value of [tex]\( x \)[/tex]
- Year 1970 is 21 years after 1949, so [tex]\( x = 21 \)[/tex].
### Step 2: Calculate the population in 1970 using the linear function [tex]\( f(x) \)[/tex]
[tex]\[ f(x) = 0.074x + 2.294 \][/tex]
Substitute [tex]\( x = 21 \)[/tex] into the function:
[tex]\[ f(21) = 0.074 \times 21 + 2.294 \][/tex]
[tex]\[ f(21) = 1.554 + 2.294 \][/tex]
[tex]\[ f(21) = 3.848 \][/tex]
### Step 3: Calculate the population in 1970 using the exponential function [tex]\( g(x) \)[/tex]
[tex]\[ g(x) = 2.577 \times 1.017^x \][/tex]
Substitute [tex]\( x = 21 \)[/tex] into the function:
[tex]\[ g(21) = 2.577 \times 1.017^{21} \][/tex]
[tex]\[ g(21) = 2.577 \times 1.425385 \][/tex]
[tex]\[ g(21) \approx 3.6715921270050353 \][/tex]
### Step 4: Compare the calculated populations with the actual population in 1970
- The actual population in 1970 was 3.7 billion.
### Step 5: Calculate the differences between the actual population and the populations predicted by each model
1. For the linear function [tex]\( f(x) \)[/tex]:
[tex]\[ \text{Difference } f = |3.848 - 3.7| \][/tex]
[tex]\[ \text{Difference } f = 0.148 \][/tex]
2. For the exponential function [tex]\( g(x) \)[/tex]:
[tex]\[ \text{Difference } g = |3.6715921270050353 - 3.7| \][/tex]
[tex]\[ \text{Difference } g \approx 0.028407872994964833 \][/tex]
### Step 6: Determine which function has the smaller difference
- The difference for [tex]\( f(x) \)[/tex] is 0.148.
- The difference for [tex]\( g(x) \)[/tex] is approximately 0.0284.
Since the difference for the exponential function [tex]\( g(x) \)[/tex] is smaller than the difference for the linear function [tex]\( f(x) \)[/tex], the exponential function [tex]\( g(x) \)[/tex] better serves as a model for the world population in 1970.
### Conclusion
The exponential function [tex]\( g \)[/tex] serves as a better model for the world population in 1970. Therefore, the answer is:
b. the exponential function [tex]\( g \)[/tex]
